Multi-channel signals captured by spatially separated sensors often contain a high level of data redundancy. A compact signal representation enables more efficient storage and processing, which has been exploited for data compression, noise reduction, and speech and image coding. This paper focuses on the compact representation of speech signals acquired by spherical microphone arrays. A polynomial matrix eigenvalue decomposition (PEVD) can spatially decorrelate signals over a range of time lags and is known to achieve optimum multi-channel data compaction. However, the complexity of PEVD algorithms scales at best cubically with the number of channel signals, e.g., the number of microphones comprised in a spherical array used for processing. In contrast, the spherical harmonic transform (SHT) provides a compact spatial representation of the 3-dimensional sound field measured by spherical microphone arrays, referred to as eigenbeam signals, at a cost that rises only quadratically with the number of microphones. Yet, the SHT's spatially orthogonal basis functions cannot completely decorrelate sound field components over a range of time lags. In this work, we propose to exploit the compact representation offered by the SHT to reduce the number of channels used for subsequent PEVD processing. In the proposed framework for signal representation, we show that the diagonality factor improves by up to 7 dB over the microphone signal representation with a significantly lower computation cost. Moreover, when applying this framework to speech enhancement and source separation, the proposed method improves metrics known as short-time objective intelligibility (STOI) and source-to-distortion ratio (SDR) by up to 0.2 and 20 dB, respectively.

Speech enhancement algorithms using polynomial matrix eigenvalue decomposition (PEVD) have been shown to be effective for noisy and reverberant speech. However, these algorithms do not scale well in complexity with the number of channels used in the processing. For a spherical microphone array sampling an order-limited sound field, the spherical harmonics provide a compact representation of the microphone signals in the form of eigenbeams. We propose a PEVD algorithm that uses only the lower dimension eigenbeams for speech enhancement at a significantly lower computation cost. The proposed algorithm is shown to significantly reduce complexity while maintaining full performance. Informal listening examples have also indicated that the processing does not introduce any noticeable artefacts.

Speech enhancement is important for applications such as telecommunications, hearing aids, automatic speech recognition and voice-controlled systems. Enhancement algorithms aim to reduce interfering noise and reverberation while minimizing any speech distortion. In this work for speech enhancement, we propose to use polynomial matrices to model the spatial, spectral and temporal correlations between the speech signals received by a microphone array and polynomial matrix eigenvalue decomposition (PEVD) to decorrelate in space, time and frequency simultaneously. We then propose a blind and unsupervised PEVD-based speech enhancement algorithm. Simulations and informal listening examples involving diverse reverberant and noisy environments have shown that our method can jointly suppress noise and reverberation, thereby achieving speech enhancement without introducing processing artefacts into the enhanced signal.

The enhancement of noisy speech is important for applications involving human-to-human interactions, such as telecommunications and hearing aids, as well as human-to-machine interactions, such as voice-controlled systems and robot audition. In this work, we focus on reverberant environments. It is shown that, by exploiting the lack of correlation between speech and the late reflections, further noise reduction can be achieved. This is verified using simulations involving actual acoustic impulse responses and noise from the ACE corpus. The simulations show that even without using a noise estimator, our proposed method simultaneously achieves noise reduction, and enhancement of speech quality and intelligibility, in reverberant environments over a wide range of SNRs. Furthermore, informal listening examples highlight that our approach does not introduce any significant processing artefacts such as musical noise.

The Second-order Sequential Best Rotation (SBR2) algorithm, used for Eigenvalue Decomposition (EVD) on para-Hermitian polynomial matrices typically encountered in wideband signal processing applications like multichannel Wiener filtering and channel coding, involves a series of delay and rotation operations to achieve diagonalisation. In this paper, we proposed the use of Householder transformations to reduce polynomial matrices to tridiagonal form before zeroing the dominant element with rotation. Similar to performing Householder reduction on conventional matrices, our method enables SBR2 to converge in fewer iterations with smaller order of polynomial matrix factors because more off-diagonal Frobenius-norm (F-norm) could be transferred to the main diagonal at every iteration. A reduction in the number of iterations by 12.35% and 0.1% improvement in reconstruction error is achievable.